- WE WANT TO DETERMINE
THE EQUATION OF A LINE THAT IS PERPENDICULAR
TO Y = 2X + 1, AND PASSES THROUGH THE POINT
WITH COORDINATES (-4,5). IF TWO LINES ARE PERPENDICULAR, THAT MEANS THEY INTERSECT
AND FORM A RIGHT ANGLE OR A 90 DEGREE ANGLE. AND IF TWO LINES
ARE PERPENDICULAR, THEN THEIR SLOPES ARE NEGATIVE
RECIPROCALS OF ONE ANOTHER, WHICH CAN BE WRITTEN
USING THIS NOTATION HERE, WHICH CAN BE
A LITTLE BIT CONFUSING. SO FOR EXAMPLE,
IF THIS FIRST LINE HAD A SLOPE OF LET'S SAY, 2/5, THEN THE SLOPE
OF THE SECOND LINE HERE WOULD HAVE TO BE
THE NEGATIVE RECIPROCAL, MEANING WE'LL FLIP THIS OVER
AND THEN CHANGE THE SIGN. SO IT'S GOING TO BE -5/2. OKAY, SO BACK TO OUR PROBLEM, WE WANT A LINE THAT IS
PERPENDICULAR TO Y = 2X + 1. WELL, THIS LINE
IS IN SLOPE INTERCEPT FORM, SO WE SHOULD RECOGNIZE THAT
THE SLOPE OF THIS LINE IS 2/1. THEREFORE, THE SLOPE
OF THE PERPENDICULAR LINE WOULD BE THE NEGATIVE RECIPROCAL
OF THIS. SO FOR THE SLOPE OF OUR LINE, WE'LL HAVE TO FLIP THIS OVER
AND CHANGE THE SIGN. THAT WOULD GIVE US -1/2. -1/2 IS A NEGATIVE RECIPROCAL
OF 2/1. AND THEN WE KNOW OUR LINE ALSO
PASSES THROUGH THE POINT (-4,5). NOW, TO DETERMINE THE EQUATION
OF THE LINE WITH SLOPE OF -1/2 CONTAINING THE POINT (-4,5), WE CAN EITHER USE
SLOPE INTERCEPT FORM, OR THE POINT SLOPE FORM
OF THE LINE. AND I THINK FOR THIS VIDEO
WE'LL SHOW BOTH. SO USING JUST SLOPE INTERCEPT
FORM OR THE FORM Y = MX + B, WE WOULD FIRST SUBSTITUTE -1/2
FOR M. SO WE'D HAVE Y = -1/2X + B. AND THEN TO DETERMINE
THE Y INTERCEPT, IT'S NOT GOING TO BE +5 BECAUSE
THE X COORDINATE IS NOT ZERO. BUT SINCE THIS POINT
IS ON THE LINE THAT MEANS IF WE SUBSTITUTE
THESE COORDINATES INTO THE EQUATION IT SHOULD SATISFY THE EQUATION. SO WE'LL SUBSTITUTE -4 FOR X,
5 FOR Y, AND THEN SOLVE FOR B. SO WE'D HAVE 5 = -1/2 x -4 + B. PUT THAT -4/1
SO WE HAVE 5 EQUALS-- AND SO THIS SIMPLIFIES HERE,
THERE'S ONE 2 IN 2, AND TWO 2s IN 4. THIS BECOMES -1 x -2. DON'T FORGET
ABOUT THE SIGN HERE. SO NOW IF WE SUBTRACT 2
ON BOTH SIDES-- WE HAVE B EQUALS--
THIS WILL BE +3. SO NOW WE HAVE OUR EQUATION, AND
WE'LL JUST SUBSTITUTE +3 FOR B. AND THE EQUATION WILL BE
Y = -1/2X + 3. AND LET'S GO AHEAD AND SEE
IF WE CAN GET THE SAME EQUATION USING POINT SLOPE FORM. POINT SLOPE FORM OF A LINE IS
Y - Y1 = M(X - X SUB 1), WHERE M IS THE SLOPE, AND (X SUB 1,Y SUB 1)
IS A POINT ON THE LINE. SO FOR THIS SITUATION
X SUB 1 IS GOING TO BE -4, AND Y SUB 1 WILL BE +5. SO WE'D HAVE Y - 5 = -1/2(X). NOW, HERE WE'D HAVE MINUS -4,
WHICH IS THE SAME AS +4. AND NOW WE'LL GO AHEAD
AND SOLVE THIS EQUATION FOR Y. SO WE'LL DISTRIBUTE HERE, SO WE'LL HAVE Y - 5 =
THIS WOULD BE -1/2X. THEN HERE WE'D HAVE -2, AND THE LAST STEP TO SOLVE FOR Y
WOULD BE TO ADD 5. THIS WOULD BE 0,
SO WE'D HAVE Y = -1/2X, AND THIS WILL BE +3. SO NOTICE HOW IF WE JUST USE
SLOPE INTERCEPT FORM TO DETERMINE THE EQUATION, OR IF WE USE POINT SLOPE FORM
TO DETERMINE THE EQUATION, THE RESULT IS THE SAME. AND LET'S GO AHEAD AND QUICKLY
VERIFY THIS GRAPHICALLY. YOU WERE GIVEN THE RED LINE
Y = 2X + 1, AND WE DETERMINED
THAT THE EQUATION OF THE PERPENDICULAR LINE
WAS Y = -1/2X + 3. NOTICE HOW THESE TWO LINES DO
INTERSECT IN A RIGHT ANGLE HERE, AND THE BLUE LINE DOES CONTAIN
THE GIVEN POINT THAT HAD A COORDINATES (-4,5),
THIS POINT HERE. SO OUR WORK LOOKS GOOD. I HOPE YOU FOUND THIS HELPFUL.